Let $\{X_{1},\ldots ,X_{n}\}$ be a sequence of dependent random variables. For $i,j \in [n]$, given
\begin{align} E[X_{i}] &= 0.\\ \text{Var}(X_{i}) &= \sigma^2 < \infty.\\ \text{Cov}(X_{i}, X_{j}) &= 0, \quad i \neq j. \end{align}
Can we say something about the asymptotic normality of $X_{1}+ \ldots+ X_{n}$ as $n \rightarrow \infty$. Any help or lead would be highly appreciated.